International Journal of Computer Discovered Mathematics (IJCDM) ISSN 2367-7775 ©IJCDM Volume 6, 2021, pp. 78–83 Received 27 February 2021. Published on- 15 April 2021 web: http://www.journal-1.eu/ ©The Author(s) This article is published with open access1.

Inequalities Involving Gergonne and Nagel Cevians

Stanley Rabinowitz 545 Elm St Unit 1, Milford, New Hampshire 03055, USA e-mail: [email protected] web: http://www.StanleyRabinowitz.com/

Abstract. A Gergonne cevian is the cevian through the Gergonne point of a . A Nagel cevian is the cevian through the Nagel point of a triangle. We present some new inequalities involving the lengths of the Gergonne and Nagel cevians of a triangle. Mathematica was used to both discover and prove some of these results. Keywords. triangle , Gergonne cevian, Nagel cevian, inequalities, Math- ematica, computer-discovered mathematics. Mathematics Subject Classification (2020). 51M04, 51-08.

1. Introduction

Let s denote the of a triangle with sides of lengths a, b, and c. Let r and R denote the inradius and circumradius of the triangle. Let ma, ha, and wa denote the length of the to side a, the to side a, and the angle bisector of angle A, respectively. Let the corresponding lengths to sides b and c be named similarly. Inequalities involving these lengths abound in the literature. See [2] and [12] for a large number of such inequalities. Typical inequalities (taken from these books) are shown below.

3 X s < m < 2s 2 a X √ ha ≤ s 3

X X X 9 (1) 9r ≤ h ≤ w ≤ m ≤ R a a a 2

1This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited. 78 Stanley Rabinowitz 79

X 2 X 2 2 X 2 (2) ha ≤ wa ≤ s ≤ ma

In this paper, we find similar inequalities involving the Gergonne and Nagel ce- vians of a triangle. (A cevian of a triangle is the line segment from a vertex to a point on the opposite side. A cevian through the Gergonne point of a triangle is called a Gergonne cevian and a cevian through the Nagel point of a triangle is called a Nagel cevian.) The lengths of the Gergonne and Nagel cevians to side a of a triangle will be called ga and na, respectively. The corresponding lengths to sides b and c are named similarly.

2. The Results

Theorem 1. The lengths of the cevians to side a of 4ABC satisfy the following chain of inequalities.

(3) 2r < ha ≤ ga ≤ wa ≤ ma ≤ na < s

Proof. The first inequality comes from [12, p. 14]. The next four inequalities come from [13]. The last inequality comes from the fact that a Nagel cevian splits the of a triangle into two equal parts. See [11, pp. 1–14]. If AN is the Nagel cevian to side a of 4ABC as shown in Figure 1, then AC + CN = s and so AN < s by the triangle inequality in 4ANC. 

Figure 1. A Nagel cevian

In this paper, we will use the summation symbol, P, to represent a cyclic sum under the mapping a → b → c. Using this notation, we have the following immediate corollary.

Corollary 2. The cyclic sums satisfy the following chain of inequalities. X X X X X ha ≤ ga ≤ wa ≤ ma ≤ na < 3s

Corollary 2 adds ga and na into the inequality chain (1). 80 Inequalities Involving Gergonne and Nagel Cevians

Since 0 < x < y implies that x2 < y2, we immediately get the following from equation (3).

Corollary 3. The lengths of the cevians to side a of 4ABC satisfy the following two chains of inequalities.

2 2 2 2 2 2 2 (4) 4r < ha ≤ ga ≤ wa ≤ ma ≤ na < s and

2 X 2 X 2 X 2 X 2 X 2 2 (5) 12r < ha ≤ ga ≤ wa ≤ ma ≤ na < 3s

We note that equality holds for an and that the first inequality in (5) is not the best possible. An improvement on the first inequality is

2 X 2 (6) 27r ≤ ha.

This result comes from [12, p. 201].

Before improving these inequalities, let us note the formulas for the lengths of various cevians of a triangle. These formulas are easily obtained using Stewart’s Theorem [1, p. 152]. Formulary. The lengths of various cevians of a triangle can be found from the following formulas. 4K2 h2 = a a2

2b2 + 2c2 − a2 m2 = a 4

4bcs(s − a) w2 = a (b + c)2

(s − a)(as − (b − c)2) g2 = a a

c2(s − b) + b2(s − c) − a(s − b)(s − c) n2 = a a where K = ps(s − a)(s − b)(s − c)

We can now look for inequalities involving the squares of the lengths of the Ger- gonne and Nagel cevians.

Theorem 4. The squares of the lengths of the Gergonne and Nagel cevians satisfy the following inequalities. 1 X X X n2 < g2 ≤ n2 4 a a a Stanley Rabinowitz 81

Proof. The second inequality comes from equation (5). The first inequality can be proven using Mathematica. The following code segment sets up the necessary variables. (The csum function creates a cyclic sum.)

assum = a>0 && b>0 && c>0 && a+b>c && b+c>a && c+a>b; csum[expr_]:= expr+(expr/.{a->b,b->c,c->a}) +(expr/.{a->c,b->a,c->b}); s = (a+b+c)/2; ga = Sqrt[(s-a)(a*s-(b-c)^2)/a]; na = Sqrt[((s-b)c^2+(s-c)b^2-a(s-b)(s-c))/a];

The command Simplify[expr,assum] instructs Mathematica to simplify the specified expression using the specified assumptions.

The following Mathematica command then proves the inequality.

Simplify[(1/4)csum[na^2]

True indicating that the inequality is true. 

If we had not known the constant 1/4 in this inequality, but merely suspected P 2 P 2 that there was some integer k such that k na < ga, we could use Mathe- matica to find the best possible value for k as follows. The Mathematica com- mand Minimize[{expr,constraint},vars] instructs Mathematica to minimize the specified expression subject to the specified constraint involving the specified variables. The Mathematica command Minimize[{csum[ga^2]/csum[na^2], assum}, {a,b,c}] produces the result {1/4, {a->1, b->1/2, c->1/2}} 1 indicating that the minimum value is 4 and that this minimum is attained when 1 1 a = 1, b = 2 , and c = 2 . The Mathematica output also includes a note that there is no minimum in the specified region and that the returned result is on the boundary of that region. This provides an alternate proof to Theorem 4. Using Mathematica in the same manner, we were able to prove the following results.

Theorem 5. The lengths of the Gergonne and Nagel cevians satisfy the following inequalities. 1 w < g ≤ w 2 a a a

ma ≤ na < 2ma 82 Inequalities Involving Gergonne and Nagel Cevians

Theorem 6. The squares of the lengths of the Gergonne cevians satisfy the fol- lowing inequalities. 1 X s2 < g2 ≤ s2 2 a 1 X 3 (a2 + b2 + c2) < g2 ≤ (a2 + b2 + c2) 3 a 4 4 X X X m2 < g2 ≤ m2 9 a a a X 27 27r2 ≤ g2 ≤ R2 a 4 √ X 2 3 3K ≤ ga Theorem 7. The squares of the lengths of the Nagel cevians satisfy the following inequalities. 2 X 2 2 s ≤ na < 3s 3 X 3 (a2 + b2 + c2) ≤ n2 < (a2 + b2 + c2) 4 a 2 X 2 X 2 X 2 ma ≤ na < 2 ma 2 X 2 2 27r ≤ na ≤ 12R √ X 2 3 3K ≤ na Theorem 8. The squares of the lengths of the Gergonne and Nagel cevians satisfy the following inequalities. X 2 X 2 2 na − ga < 2s 3 X 1 X 11 s2 ≤ n2 − g2 < s2 4 a 4 a 4 For additional inequalities involving Gergonne and Nagel cevians, see references [3, 4, 5, 6, 7, 8, 9, 10].

References [1] Nathan Altshiller-Court, College Geometry, 2nd edition, Barnes & Noble, New York, 1952. [2] Oene Bottema, R. Z.ˇ Djordjevi´c,R. R. Janic, Dragoslav S. Mitrinovi´c,and P. M. Vasic, Geometric Inequalities, Wolters-Noordhoff Publishing, Groningen, The Netherlands, 1969. [3] Bogdan Fustei, About Nagel and Gergonne’s Cevians, Romanian Mathematical Magazine, 2019. http://www.ssmrmh.ro/wp-content/uploads/2019/07/ ABOUT-NAGEL-AND-GERGONNE’S-CEVIANS.pdf [4] Bogdan Fustei, About Nagel and Gergonne’s Cevians (II), Romanian Mathematical Magazine, 2019. http://www.ssmrmh.ro/wp-content/uploads/2019/10/ ABOUT-NAGELS-AND-GERGONNES-CEVIANS-_II_.pdf [5] Bogdan Fustei, About Nagel and Gergonne’s Cevians (III), Romanian Mathematical Magazine, 2020. http://www.ssmrmh.ro/wp-content/uploads/2020/02/ ABOUT-NAGELS-AND-GERGONNES-CEVIANS-III.pdf [6] Bogdan Fustei, About Nagel and Gergonne’s Cevians (IV), Romanian Mathematical Magazine, 2020. http://www.ssmrmh.ro/wp-content/uploads/2020/02/ ABOUT-NAGELS-AND-GERGONNES-CEVIANS-_IV_.pdf [7] Bogdan Fustei, About Nagel and Gergonne’s Cevians (V), Romanian Mathematical Magazine, 2020. http://www.ssmrmh.ro/wp-content/uploads/2020/06/ ABOUT-NAGELS-AND-GERGONNES-CEVIANS-V.pdf Stanley Rabinowitz 83

[8] Bogdan Fustei, About Nagel and Gergonne’s Cevians (VI), Romanian Mathematical Magazine, 2020. http://www.ssmrmh.ro/wp-content/uploads/2020/12/ ABOUT-NAGELS-AND-GERGONNES-CEVIANS-VI.pdf [9] Bogdan Fustei, About Nagel and Gergonne’s Cevians (VII), Romanian Mathematical Magazine, 2021. http://www.ssmrmh.ro/wp-content/uploads/2021/01/ ABOUT-NAGELS-AND-GERGONNES-CEVIANS-VII.pdf [10] Bogdan Fustei, About Nagel and Gergonne’s Cevians (VIII), Romanian Mathematical Magazine, 2021. http://www.ssmrmh.ro/wp-content/uploads/2021/01/ ABOUT-NAGELS-AND-GERGONNES-CEVIANS-VIII.pdf [11] Ross Honsberger, Cleavers and Splitters. Chapter 1 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Mathematical Association of America, Washington, DC, 1995. [12] Dragoslav S. Mitrinovi´c,J. Peˇcari´c,and Vladimir Volenec. Recent advances in geometric inequalities. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989. [13] Stanley Rabinowitz, Problem 4245, School Science and Mathematics, 89(1989)445.